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XLeratorDB/financial-options Documentation

SQL Server Binomial Tree function for American option implied volatility


BinomialAmericanIV

Updated: 04 Sep 2012


Use BinomialAmericanIV to calculate the implied volatility of an American option using the Binomial Tree option pricing formula.
Syntax
SELECT [wctOptions].[wct].[BinomialAmericanIV] (
  <@CallPut, nvarchar(4000),>
 ,<@AssetPrice, float,>
 ,<@StrikePrice, float,>
 ,<@TimeToMaturity, float,>
 ,<@RiskFreeRate, float,>
 ,<@DividendRate, float,>
 ,<@Price, float,>
 ,<@NSteps, int,>)
Arguments
@CallPut
identifies the option as being a call ('C') or a put ('P'). @CallPut is an expression of type nvarchar or of a type that can be implicitly converted to nvarchar.
@AssetPrice
the price of the underlying asset. @AssetPrice is an expression of type float or of a type that can be implicitly converted to float.
@StrikePrice
the exercise price of the option. @StrikePrice is an expression of type float or of a type that can be implicitly converted to float.
@TimeToMaturity
the time to expiration of the option, expressed in years. @TimeToMaturity is an expression of type float or of a type that can be implicitly converted to float.
@RiskFreeRate
the annualized, continuously compounded risk-free rate of return over the life of the option. @RiskFreeRate is an expression of type float or of a type that can be implicitly converted to float.
@DividendRate
the annualized, continuously compounded dividend rate over the life of the option. For currency options, @DividendRate should be the foreign risk-free interest rate. @DividendRate is an expression of type float or of a type that can be implicitly converted to float.
@Price
the observed price of the option. @Price is an expression of type float or of a type that can be implicitly converted to float.
@nSteps
the number of steps in the binomial tree. @nSteps is an expression of type int or of a type that can be implicitly converted to int.
Return Type
float
Remarks
·         @TimeToMaturity must be greater than zero (@TimeToMaturity > 0).
·         @AssetPrice must be greater than zero (@AssetPrice > 0).
·         @StrikePrice must be greater than zero (@StrikePrice > 0).
·         @nSteps must be greater than 1 (@nSteps > 1)
·         If @DividendRate is NULL an error will be returned
·         If @RiskFreeRate is NULL an error will be returned
·         To calculate the price or Greeks of an American option use BjerksundStensland or BinomialAmerican.
·         To calculate the implied volatility of an American option using the Bjerksund & Stensland 2002 formula, use BjerksundStenslandIV.
·         To calculate the implied volatility on European options use BlackScholesMertonIV or BinomialEuroIV.
Example
Calculate the price for a call option on 2012-09-04, expiring on 2012-12-15, with a current asset price of 99.5, a strike price of 100 and a volatility of 20%. The observed price is 4.1547 and the dividend rate is 0.5%. The number of steps is 100.
SELECT ROUND(wct.BinomialAmericanIV(
      'C'         --PutCall
      ,99.5       --Asset Price
      ,100        --Strike Price
      ,datediff(d,'2012-09-04','2012-12-15') / 365.0000     --Time-to-expiry
      ,.02        --Risk Free Rate
      ,.005       --Dividend Rate
      ,4.1547           --Price
      ,100        --Number of Steps
      ), 5) as Volatility
This produces the following result.
            Volatility
----------------------
                   0.2
 
In the following SELECT, we collect the implied volatility for a series of strike prices on EUR/USD option which expire in 7 days time. The option price is the midpoint of the bid and ask and the 7 day EURIBOR rate is 0.0335700% and the 7 day LIBOR rate is 0.1870000%.
SELECT
CASE
      WHEN call_bid IS NULL THEN NULL
      WHEN call_ask IS NULL THEN NULL
      ELSE wct.BinomialAmericanIV('C'
      ,127.91           --Asset price    
      ,n.s              --Strike   
      ,Cast(7 as float)/cast(365 as float) --Time
      ,0.000335699      --Continuously Compouned EURIBOR
      ,0.001869966      --Continuously Compouned LIBOR
      ,(call_bid + call_ask) / 2    --Price
      ,100              --Number of Steps
      )
 END as [Call IV]
,cast(n.call_bid as money) as Bid
,cast(n.call_ask as money) as Ask
,n.s as Strike
,cast(n.put_bid as money) as Bid
,cast(n.put_ask as money) as Ask
,CASE
      WHEN put_bid IS NULL THEN NULL
      WHEN put_ask IS NULL THEN NULL
      ELSE wct.BinomialAmericanIV('P'
      ,127.91           --Asset price
      ,n.s              --Strike
      ,Cast(7 as float)/cast(365 as float)      --Time
      ,0.000335699      --Continuously Compouned EURIBOR
      ,0.001869966      --Continuously Compouned LIBIBOR
      ,(put_bid + put_ask) / 2      --Price
      ,100              --Number of Steps
      )
 END as [Put IV]
FROM (
      SELECT 6.85, 7.00, 121, NULL, .04 UNION ALL
      SELECT 5.85, 6.00, 122, NULL, .04 UNION ALL
      SELECT 4.85, 5.00, 123, NULL, .04 UNION ALL
      SELECT 3.85, 4.00, 124, NULL, .04 UNION ALL
      SELECT 2.91, 2.99, 125, .02, .05 UNION ALL
      SELECT 2.00, 2.06, 126, .10, .13 UNION ALL
      SELECT 1.20, 1.25, 127, .29, .32 UNION ALL
      SELECT 0.59, 0.62, 128, .68, .71 UNION ALL
      SELECT 0.23, 0.25, 129, 1.31, 1.35 UNION ALL
      SELECT 0.06, 0.09, 130, 2.13, 2.19 UNION ALL
      SELECT 0.01, .04, 131, 3.05, 3.15
      ) n(call_bid, call_ask, s, put_bid, put_ask)
 
This produces the following result.
XLeratorDB function example BinomialAmericanIV for SQL Server 


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